# Contributions to Probability

### A Collection of Papers Dedicated to Eugene Lukacs

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Contributions to Probability: A Collection of Papers Dedicated to Eugene Lukacs is a collection of papers that reflect Professor Eugene Lukacs’ broad range of research interests. This text celebrates the 75th birthday of Eugene Lukacs, mathematician, teacher, and research worker in probability and mathematical statistics. This book is organized into two parts encompassing 23 chapters. Part I consists of papers in probability theory, limit theorems, and stochastic processes. This part also deals with the continuation and arithmetic of distribution functions, the arc sine law, Fourier transform methods, and nondifferentiality of the Wiener sheet. Part II includes papers in information and statistical theories. This book will prove useful to statisticians, mathematicians, and advance mathematics students.

## Table of Contents

List of Contributors

Preface

Eugene Lukacs

Part I. Probability

Probability Theory

Lagrange's Theorem and Thin Subsequences of Squares

References

A Kind of Random (= Stochastic) Integral

0. Random Functions

1. Measurable Random Functions

2. Random ( = Stochastic) Measure

3. Random Integral

References

Continuation of Distribution Functions

1. Introduction

2. The Normal Distribution Function

3. Analytic Distribution Functions

4. Continuation Theorems for Special Classes of i.d. d.f.

5. Uniqueness of Symmetric Distribution Functions

References

The Arc Sine Law of Paul Lévy

1. Introduction

2. The Coin Tossing Game

3. Paul Lévy's Arc Sine Law

4. An Alternative Approach

5. Exact Distributions

6. Notes

7. Lévy's Heuristic Method

8. The Case of Independent Sequences

9. The Case of Dependent Sequences

References

Limit Theorems

General Limit Theorems for Products with Applications to Convolution Products of Measures

1. Introduction

2. Some Fundamental Identities and Inequalities

3. Convergence of Powers in Seminorms to Infinitely Divisible Elements

4. Convergence of Products in Seminorms

References

Stable Limit Law and Weak Law of Large Numbers for Hilbert Space with "Large-O" Rates

1. Introduction

2. Notations and Preliminaries

3. Two General Large-O Approximation Theorems

4. Stable Limit Law on H with Rates

5. The Central Limit Theorem on H

6. The Weak Law of Large Numbers

7. Limit Theorems for Random Vectors in ℝm

References

The Arithmetic of Distribution Functions

1. Introduction

2. The Class L

3. Infinitely Divisible Characteristic Functions with Absolutely Continuous Spectral Functions

4. Products of Poisson-Type Characteristic Functions

5. Independent Sets

6. Infinitely Divisible Characteristic Functions with Continuous Spectral Functions

7. Indecomposable Laws

8. Indecomposable Factors

9. α Decompositions

References

On the Tails of a Class of Infinitely Divisible Distributions

1. Introduction

2. Some Lemmas

3. Proof of Theorem 1.2

4. On Another Theorem of Elliott and Erdös

References

Fourier Transform Methods in the Study of Limit Theorems in a Hilbert Space

1. Introduction

2. Preliminaries

3. Infinitely Divisible Probability Measures

4. The General Central Limit Problem in a Hilbert Space

5. Self-Decomposable and Stable Measures in ℋ

6. Semi-Stable Measures in ℋ

References

Polynomials in Gaussian Variables and Infinite Divisibility?

References

On the Nondifferentiability of the Wiener Sheet

1. Introduction

2. The Nondifferentiability of the Wiener Sheet in the Direction of the X Axis

3. The Wiener Sheet Is Nowhere Differentiable in Any Direction

References

The Degree of Vertices on a Randomly Growing Tree

1. Introduction

2. The State-Homogeneous Case

3. A Simple Practical Example

References

Uniform Convergence of Random Trigonometric Series and Sample Continuity of Weakly Stationary Processes

1. Introduction

2. Sequences of Random Variables Satisfying the Condition Mr

3. Uniform Convergence of Random Trigonometric Series

4. Weakly Stationary Process of Class Mr: Approximate Fourier Series

5. Sample Continuity of a Weakly Stationary Process of Mr

References

Stochastic Equations Driven by Random Measures and Semimartingales

Introduction

Notations

1. Some Preliminaries on Random Measures

2. Stochastic Equations Driven by White Random Measures

3. Equations Driven by Random Measures and Semimartingales

References

Part II. Applications of Probability

Information Theory

Derivations and Information Functions (A Tale of Two Surprises and a Half)

1. Introduction

2. General and Regular Information Functions

3. Nonnegative Information Functions

4. Solutions of (12)

References

Entropy of the Sum of Independent Bernoulli Random Variables and of the Multinomial Distribution

1. Introduction

2. Entropy of Sum of Independent Bernoulli Random Variables

3. Entropy of the Multinomial Distribution

References

On the Concept and Measure of Information Contained in an Observation

Introduction

1. Derivation of a Measure for the Information Contained in an Observation

2. The Average Information

3. Fisher Information, as Information Density

4. Connection with the Variance Bound of the Cramér-Fréchet-Rao Inequality

5. The Bayesian Approach

6. Can the Measure of Information be a Distance in the Parameter Space?

References

Statistical Theory

The Empirical Characteristic Process When Parameters Are Estimated

1. Introduction

2. Notation

3. Assumptions

4. The Integrated Squared Error Estimator

5. The Integrated Error Estimator

6. The Parameter Estimated Empirical Characteristic Process with the Integrated Squared Error Estimator

7. The Parameter Estimated Empirical Characteristic Process with the Integrated Error Estimator

8. Remarks

References

Identifiability

1. Introduction

2. Identification in Errors in the Variables Models

3. Identifiability of ARMA Systems

References

On a Multivariate Extension of the Behrens - Fisher Law

References

Constant Regression of Quadratic Statistics on the Sum of Random Variables Defined on a Markov Chain

1. Introduction

2. Preliminaries

3. On the Constant Regression of Quadratic Forms on the Sum of Random Variables Defined on a Markov Chain

4. The Characterization of the Normal Law by the Constant Regression of Quadratic Statistics on the Sum of Random Variables Defined on a Markov Chain

References

Invariance Principles for Rank Statistics for Testing Independence

1. Introduction

2. Assumptions and Some Preliminary Lemmas

3. Order of Magnitude of the Remainder Term

4. Invariance Principles

References

Integral Transformations of Distributions and Estimates of Parameters of Multidimensional Spherically Symmetric Stable Laws

1. Introduction

2. Some Properties of Spherically Symmetric Distributions

3. Integral Transformations of Distributions

4. The Estimation of Parameters of Spherically Symmetric Stable Laws

5. Comments

References

Eugene Lukacs: Bibliography

## Product details

- No. of pages: 334
- Language: English
- Copyright: © Academic Press 1981
- Published: January 1, 1981
- Imprint: Academic Press
- eBook ISBN: 9781483262567